Multifractal analysis for Bedford-McMullen carpets
Thomas Jordan, Michal Rams
本文计算了Bedford-McMullen引入的自仿射地毯上支撑的Bernoulli测度的局部维数的多重分形分析。这扩展了King的工作，其中多重分形分析是用强附加分离假设来计算的。
In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford-McMullen. This extends the work of King where the multifractal analysis is computed with strong additional separation assumptions.
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We describe the shrinking target set for the Bedford-McMullen carpets, with targets being either cylinders or geometric balls.
Quantitative recurrence problem on some Bedford-McMullen carpets
针对 Hausdorff 维数与盒维数相等的 Bedford-McMullen 地毯，论文研究其典范自同态定量回归集的 Hausdorff 维数。
Point-wise doubling indices of measures and its application to bi-Lipschitz classification of Bedford-McMullen carpets
本文引入逐点加倍指数以刻画非加倍测度，计算了Bedford-McMullen地毯上均匀Bernoulli测度的该指数，并据此证明除一小类外，双Lipschitz等价的地毯具有相同纤维序列（至多相差一个置换）。
Packing spectra for Bernoulli measures supported on Bedford-McMullen carpets
本文考虑Bedford-McMullen地毯上Bernoulli测度的局部维数的Packing谱。我们表明，通常的包装尺寸的正规集是小于包装尺寸的吸引子。我们还考虑了一类特定的措施，我们能够准确地计算包装谱，我们表明，包装谱是不连续的伯努利措施的空间上的函数。
Weighted topological entropy and intersecting random translates of Bedford--McMullen carpets
通过建立Feng-Huang加权拓扑熵的相对变分原理，结合Yin的定理，证明了纤维上Feng-Huang熵与其组合版本的几乎处处等价，并应用于计算两个Bedford-McMullen地毯随机平移交集的Hausdorff维数，将Kenyon-Peres公式从自相似推广到自仿射情形。
In 1984 Bedford and McMullen independently introduced a family of self-affine sets now known as \emph{Bedford-McMullen carpets}. Their work stimulated a lot of research in the areas of fractal geometry and non-conformal dynamics. In this survey article we discuss some aspects of Bedford-McMullen carpets, focusing mostly on dimension theory.
In this paper, we study the topological properties and the gap sequences of Bedford-McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford-McMullen sets and prove that all normal Bedford-McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymp…
Mean dimension theory for infinite dimensional Bedford-McMullen sponges
本文针对$r\geq3$的Bedford-McMullen海绵系统建立平均维数理论，明确计算其度量平均维数与平均豪斯多夫维数，扩展了Tsukamoto的相关结果，并给出两者相等的特殊情形。
Hausdorff dimension of reflected Bedford--McMullen carpets
研究反射贝德-麦克穆伦地毯的豪斯多夫维度，通过弱坐标投影熵控制维度，推导出显式公式并讨论反射模式对投影熵的影响。
Let $F \subseteq \mathbb{R}^2$ be a Bedford-McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between $0$ and $1$. We prove that any similarity $g$ such that $g(F) \subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by "zooming in" on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self similar se…
The intermediate dimensions of a set $Λ$, elsewhere denoted by $\dim_θΛ$, interpolates between its Hausdorff and box dimensions using the parameter $θ\in[0,1]$. Determining a precise formula for $\dim_θΛ$ is particularly challenging when $Λ$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this direction, answering a question of Fraser, we show that $\dim_θΛ$ is strictly less than the box dimension of $Λ$ for every $θ<1$, moreover, the derivative of the upper bound is strictly positive at $θ=1$. We also improve on the lower bound obtained by Falconer, Fraser and Kempt…
Positive-measure intersections of Bedford-McMullen carpets under similitudes
本文证明Bedford-McMullen地毯在相似变换下若交集有正测度，则正交部分必为对角或反对角，且等距情形下仅差平移，加强了2010年猜想。
Dimensions of Bedford-McMullen type sets in $\mathbb Z^2$
本文研究 $\mathbb Z^2$ 中由扩张仿射映射生成的 Bedford-McMullen 型前向轨道的各种维数，给出显式公式，并揭示其与半群增长及对偶吸引子维数的关系。
On the dimensions of a family of overlapping self-affine carpets
我们考虑一族与Bedford-McMullen地毯有关的自仿射集的维数。特别是，我们固定的Bedford-McMullen系统，然后随机化的列结构保持的规定下的平移向量。因此，我们保留了Bedford-McMullen设置中的一个关键特征，即对齐会导致维度从亲和维度中下降。我们计算的豪斯多夫，包装和盒尺寸以外的一个小的一组特殊的翻译，并为一些明确的翻译，即使在重叠的存在。我们的结果依赖于，并且可以被看作是，M。Hochman最近关于自相似集和测度的维数的工作。
Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence
本文利用大偏差速率函数给出了 Bedford-McMullen 地毯中间维数的精确公式，揭示其可数多个相变及严格凹性，并证明中间维数相等等价于均匀 Bernoulli 测度的 Hausdorff 多重分形谱相等，从而为 Lipschitz 等价性提供必要条件。
The Hausdorff dimension of the projections of self-affine carpets
研究了一类自仿射地毯的正交投影，其中Bedford地毯和McMullen地毯是其特例。我们的主要结果是，如果$Λ$是这样的地毯，并且某些自然非理性条件成立，则$Λ$在非主方向上的每个正交投影都有豪斯多夫维数$\min(γ,1)$，其中$γ$是$Λ$的豪斯多夫维数.这推广了Peres和Shmerkin关于Cantor集和的一个结果。
We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_λ(z)=z^m+λ/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m<1$ and $λ\in\mathbb{C}\setminus\{0\}$. If the free critical points of $f_λ$ are escaped to the infinity, we prove that the Julia set $J_λ$ of $f_λ$ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a sufficient condition on $λ$ such that $J_λ$ is a Sierpiński carpet and pro…
Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford-McMullen carpets, that is, carpets $F$ and $E$ such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self affine measures on them a…
Dimensions of compact invariant sets of some expanding maps
研究了由整数值对角矩阵给出的环面上扩张非共形映射的紧不变集的豪斯多夫维数和满豪斯多夫维数测度. McMullen和Bedford发现了“一般Sierpinski地毯”的豪斯多夫维数，Kenyon和Peres证明了满豪斯多夫维数的测度在某些情况下是唯一的.我们推广这些结果，通过使用补偿函数来研究一般的Sierpinski地毯表示的移位的有限型。给出了一般Sierpinski地毯存在唯一的满豪斯多夫维数测度的条件，并研究了唯一测度的性质.
We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let $μ_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum and the doubling property of $μ_E$ are both invariant under a bi-Lipschitz map. Moreover, we show that if $μ_E$ and $μ_F$ are doubling, then a bi-Lipschitz map between $E$ and $F$ enjoys a certain measure preserving property.
Critical Hausdorff Measure for Higher-Dimensional Bedford-McMullen Subshifts
本文研究高维Bedford-McMullen子移位的临界Hausdorff测度，证明维数相等当且仅当临界测度为正且有限，并解决Peres猜想，推广至sofic子移位并构造两类反例。
Let $F\subseteq [0,1]^2$ be a Bedford-McMullen carpet defined by exponents $m>n$, that projects to $[0,1]$ on the $y$-axis. We show that under mild conditions on $F$, there are many non principle lines $\ell$ such that $\dim^* F\cap \ell = \dim^* F -1$, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). This exhibits the sharpness of recent Furstenberg-type slicing theorems obtained by Algom (2020) about upper bounds on the dimension of every such slice.
Let $F$ be a Bedford-McMullen carpet defined by independent integer exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not parallel to the major axes, $$ \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ and $$ \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ where $\dim^*$ is Furstenberg's star dimension (maximal dimension of microsets). This improves the state of art results on Furstenberg type slicing Theorems for affine invariant carpets.
We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Barański carpets. More specifically, we fix a Barański system and allow both vertical and horizontal random translations, while preserving the rows and columns structure. The alignment kept in the construction allows us to give expressions for these fractal dimensions outside of a small set of exceptional translations. Such formulas will coincide with those for the non-overlapping case, and thus provide examples where the box-counting and Hausdorff dimension do not necessarily agree. The…
We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. In particular, for self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both …
This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen c…
Hausdorff dimension of certain random self--affine fractals
在这项工作中，我们感兴趣的自仿射分形研究的Gatzouras和Lalley和作者推广了著名的一般Sierpinski地毯研究的贝德福德和McMullen。我们给出了一个公式的豪斯多夫维数的集合是随机产生的使用有限数量的自仿射变换每一个产生一个分形集如前所述，与一些技术假设。变换的选择根据伯努利测度是随机的。该公式是根据维数的变分原理给出的。
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2)…
Dimension of self-affine sets with holes
本文计算了一类动态定义的非共形集的维数。设$X\subseteq\mathbb{T}^2$表示Bedford-McMullen集，$T:X\to X$表示使$X$保持不变的自然扩张环面自同态.对于开集$U\subset X$，我们令X_U={x \in X：T^k（x）\not \in U \text{for all} k}。我们研究了$X_U$的盒维数和豪斯多夫维数，对于固定的马尔可夫洞和当$U$是一个收缩的度量球。我们证明了盒维数是由最大熵测度通过$U$的逃逸率控制的，而Hausdorff维数取决于最大维数测度的逃逸率.
Local structure of self-affine sets
具有开集条件的自相似集在放大条件下结构不变。对于自仿射集，情况完全不同。我们考虑贝德福德、麦克马伦、加佐拉斯和拉利研究的平面自仿射康托集E，它在水平轴上的投影是一个区间。我们证明了在E中几乎每个点x的正方形邻域内，对于地址空间上的许多积测度，E可以由一个区间和一个康托尔集的积集很好地逼近。即使E是全不连通的，其极限集也具有区间纤维的乘积结构，类似于混沌可微动力系统的吸引子.
The Assouad spectrum of random self-affine carpets
本文研究随机自仿贝德福德-麦克马伦地毯，推导其几乎必然Assouad谱和拟Assouad维数，发现非共形情形下它们通常不同于盒维数或Assouad维数，并给出示例说明差异。
We study the Assouad and lower dimensions of self-affine sponges; the higher dimensional analogue of the planar self-affine carpets of Bedford and McMullen. Our techniques involve the weak tangents of Mackay and Tyson as well as regularity properties of doubling measures in the context studied by Bylund and Gudayol.
On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets
研究了某些仿射不变集的豪斯多夫维数和盒维数相等性，证明了在s≤2时三者等价，且在s≥3时发现新的现象，通过概率方法建立了贝福德-麦克穆伦海绵体的等价性。
Scaling scenery of $(\times m,\times n)$ invariant measures
本文研究了非共形环面自同态$(x,y) \mapsto (mx \mod 1,ny \mod 1)$的不变测度的标度风景和极限几何，这些测度是自然马尔可夫分割的Bernoulli测度。我们表明，统计的标度可以描述的遍历CP-链的意义上的Furstenberg。对CP-链的构造进行简化，得到了伯努利测度的投影定理，它部分地推广了Hochman-Shmerkin和Ferguson-Jordan-Shmerkin的早期结果。我们还给出了一个遍历的理论准则的Falconer的距离集猜想的维数部分的一般集具有正的长度使用CP-链，从而验证了它的各种类的分形，如自仿射地毯的Bedford-McMullen，Lalley-Gatzouras和Baraavorski类和所有平面自相似集。
In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, in…
Improved dimension theory of sofic self-affine fractals
该研究建立了任意欧氏空间中自仿射分形Hausdorff维数的组合表达式，推广了平面sofic集相关研究，给出高维sofic集精确维数及平面两类维数相等的条件，同时补充了相关先行研究说明与后续工作进展。
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